Heat the oil in a large skillet over medium heat. Add onions, carrots, celery, potatoes, sausage, paprika, garlic, red pepper flakes, and salt and pepper, to taste. Saute stirring frequently, until the potatoes and carrots are tender, and the sausage is cooked through, about 12 minutes.
Bring a medium pot of water to a simmer over low heat and add the vinegar. Crack an egg into a small bowl and carefully slide it into the simmering water. Quickly repeat with remaining eggs. Using a slotted spoon, carefully corral the white of each egg around its yolk. Poach the eggs until the whites are firm, or to the desired degree of doneness, about 3 to 5 minutes.
Divide the hash among 4 serving plates and top the hash with a poached egg. Season with salt and pepper, to taste, and serve.